The L^p-norms of the Beurling–Ahlfors transform on radial functions

Abstract

Abstract. We calculate the norms of the operators connected to the action of the BeurlingAhlfors transform on radial function subspaces introduced by Bañuelos and Janakiraman. In particular, we find the norm of the Beurling-Ahlfors transform acting on radial functions for p > 2, extending the results obtained by Bañuelos and Janakiraman, Bañuelos and Osȩkowski, and Volberg for 1 < p ≤ 2.

Weighted inequalities involving Hardy and Copson operators

“…Apart from the motivation described above, there seems to be a rapidly growing interest in finding optimal bounds for inequalities of the form (1.4), both for nonnegative functions and for non-increasing functions, motivated by some very interesting connections to Poincaré or Sobolev inequalities, rearrangement estimates of BMO functions, the conjecture of Iwaniec concerning the norm of the Beurling operator, and more. The interested reader is kindly referred to checking the papers [11,12,13,27,31,32,33,36,48,49], and also the references given therein.…”

Weak-type and end-point norm estimates for Hardy operators

“…then the norm of B on such radial functions is equivalent to the norm of S − Id for general functions f . In this case, optimal constants were obtained in [4], if 1 < p ≤ 2, and in [22], for the whole range 1 < p < ∞ (see also [23] for a recent extension to power weight estimates in L p (t α ), with α < p − 1).…”

Hardy's operator minus identity and power weights

“…The case of positive decreasing functions is obviously connected to the concept of monotone rearrangements and is motivated by results about normability and embeddings of function spaces (see, e.g., [4,9,8]). On the other hand, the case of general functions in L p ([0, ∞)) corresponds to the study of the Beurling-Ahlfors transform acting on radial functions (see, e.g., [2,15] and the references therein) and is interesting even in the unweighted setting. Before presenting the contribution of the current paper, let us discuss relevant prior results.…”

“…alternative proofs were given in [3] and [16]. The best constant in the complementary range p > 2 was found by the author in [15]: the expression is more complicated and involves the root of an equation. The proof in [2] is based on properties of stretch functions, while [3,16,15] exploit ideas connected to the Bellman function technique and Burkholder's work on martingale inequalities.…”

“…for the whole range of admissible parameters p and a, when C is one of the following classes: positive decreasing functions, positive functions, general functions. To this end we adapt the techniques of [15]. This enables us to provide a uniform framework for all inequalities and kill several birds with one stone.…”